Magnetic moments induce strong phonon renormalization in FeSi

The interactions of electronic, spin and lattice degrees of freedom in solids result in complex phase diagrams, new emergent phenomena and technical applications. While electron–phonon coupling is well understood, and interactions between spin and electronic excitations are intensely investigated, only little is known about the dynamic interactions between spin and lattice excitations. Noncentrosymmetric FeSi is known to undergo with increasing temperature a crossover from insulating to metallic behaviour with concomitant magnetic fluctuations, and exhibits strongly temperature-dependent phonon energies. Here we show by detailed inelastic neutron-scattering measurements and ab initio calculations that the phonon renormalization in FeSi is linked to its unconventional magnetic properties. Electronic states mediating conventional electron–phonon coupling are only activated in the presence of strong magnetic fluctuations. Furthermore, phonons entailing strongly varying Fe–Fe distances are damped via dynamic coupling to the temperature-induced magnetic moments, highlighting FeSi as a material with direct spin–phonon coupling and multiple interaction paths.

Electrical resistivity measured on another small piece of our FeSi sample (solid line). The dashed line denotes results from Ref. 3 , which were scaled to fit our data at T = 300 K. (c) Larmor diffraction on the (2,2,2) Bragg reflection: Data are measured in the parallel field mode, which is sensitive to the spread of d-spacings and insensitive to the mosaicity. Raw data are shown in blue, whereas green data show the results normalized to the resolution measured with a (perfect) Ge crystal. The green curve is flat, which means that we see no broadening. The fit gives . This error bar is reasonable, such that the spread ⁄ . (d) Rocking scan of the (2,0,0) Bragg reflection on the 1T TAS. The scan is resolution limited, i.e. the mosaic is smaller than 0.1°.

Supplementary Note 1 Polarized neutron scattering
The paramagnetic scattering in FeSi up to energy transfers of 80 meV was studied with polarized thermal neutrons on the TAS IN20 at ILL, Grenoble. Polarizing Heusler monochromator and analyzer have been used. The paramagnetic scattering intensities can be separated from other scattering processes by taking the difference of the measured spin-flip scattering for two different orientations of the guide field present at the sample position, i.e. horizontal field (HF) and vertical field (VF) 7 , By setting the analyzer to a fixed final energy of 35 meV, energy transfers up to 80 meV could be achieved. In this study we focused on measuring the paramagnetic fluctuations close to reciprocal lattice vectors with large ferromagnetic structure factors, i.e. (1+ 1+ 0+ ), (1+ , 1 1 ), (1+ , 1 2 ) and . Magnetic scattering is strongest at small momentum transfers. However, we needed the larger wave vector values in order to close the scattering triangle at energies . Hence, we performed the scans shown in Fig. 3(b) actually in scans at different wave vectors with different energy ranges. The obtained intensities were corrected for the ferromagnetic structure factors of the reciprocal lattice points being and for and , respectively. Supplementary Figure 4(a) shows the corresponding three data sets for the measurement at T = 300 K. The wave vector dependence of magnetic scattering in FeSi has been studied using polarized neutrons in the 1980s 4,5 . Here, we re-analyze the reported data in order to extract the experimentally observed magnetic correlation length . The dynamic susceptibility reads , with the static susceptibility. is a damping parameter and denotes the correlation length. For small energy transfers, i.e., , the elastic scattering can be expressed as ( ) . Fitting this expression to published data on paramagnetic scattering at zero energy transfer 5 [ Supplementary   Figure 4(b)] we extract a magnetic correlation length in FeSi at T = 300 K of , i.e., 80% of the observed lattice parameter. The distance from an Fe ion to its 6 nearest Fe neighbors is . Hence, on a local scale probed by the atomic movements within a phonon pattern, FeSi indeed can be regarded as a ferromagnet at elevated temperatures.

Ab-initio calculations based on density-functional-perturbation theory (DFPT) Phonon energies in the quasi-harmonic approximation
A small softening of phonon energies with increasing temperature can be expected due to thermal expansion of the lattice. The Grüneisen parameter relates the strength of the softening with the change in the volume of the unit cell. If averaged over all energies and wave vectors, it acquires typically values of 1.5 -2. In principle, however, the Grüneisen parameter depends on the energy and polarization of a phonon mode. In order to have a mode-selective estimate of the phonon softening due to thermal expansion, we performed DFPT calculations with lattice constant values reported for temperatures T = 10 K, 100 K, 200 K, 300 K, 400 K, 500 K and 600 K 6 . Results plotted in Supplementary Figure 5(c) demonstrate that the expected relative softening due to thermal expansion is different for different modes. For instance, a high-energy mode at 54 meV at the R point shows a relative softening of only 3%, whereas the R1 mode energy is reduced by 6%. We note that the former is in good agreement with a 2% increase in the unit cell volume and a Grüneisen parameter of 1.5 as suggested in Ref. 8 . Obviously, this is different for the R1 mode.
This approach of using DFPT in order to estimate the phonon energies as function of temperature is known as the quasi-harmonic approximation and was validated in previous work [9][10][11] .

Calculations for metallic FeSi
FeSi has an insulating ground state and our calculations of the electronic structure show indeed a strong suppression of electronic states at E F , reminiscent of a gap in the excitation spectrum [ Supplementary Figure 5(a)]. The charge gap was reported to close around room temperature 12 and, hence, a coupling of phonons to electronic states near E F becomes possible.
DFPT is widely used to calculate the electronic contribution to the line widths of each phonon mode as function of energy and momentum in metallic compounds like charge-density-wave materials 13,14 or conventional superconductors above their respective transition temperatures 15,16 . DFPT calculations require a numerical smearing σ of the electronic bands due to the finite momentum mesh used. It was noted that this smearing can simulate a temperature effect [17][18][19] . Using in our calculations for FeSi effectively smears out all indications for a gap in the electronic excitation spectrum and yields a metallic like electronic density of states, eDOS [Supplementary Figure 5(a)]. The resulting metallic eDOS looks very similar to calculations by Jarlborg et al. 20 Based on this electronic structure it is possible to investigate the above discussed electron-phonon-coupling in the high-temperature, metallic phase of FeSi. We calculated the line widths of individual modes based on the electron-phonon coupling. Here, a denser 16x16x16 k-point mesh was employed for better convergence. Our calculations yield electronic contributions to the phonon line widths of no more than 0.1 meV for any mode in FeSi.

Spin polarized phonon calculations
Our experimental results suggest a close relationship between the increase in the phonon line widths of the R1 and R2 modes as a function of temperature with the temperature dependent magnetic moment in FeSi. The effect of a magnetic ground state on the lattice dynamical properties can be estimated employing spin polarized DFPT calculations. This has been done, e.g., for the Fe-based superconductors and related compounds 21,22 as well as for MnSi 23 . For the experimental lattice constant of FeSi, i.e. a = 4.48 Å, spin-polarized calculations show that a magnetic state is not stable. However, we found that increasing the lattice constant to a* = 4.65 Å results in a stable magnetic ground state for FeSi. Using a larger lattice constant was further motivated by the presence of magnetic order in isostructural FeGe having a lattice constant of 4.7 Å. 24 FeGe features a helical magnetic order with an extremely long pitch of the helix of 700 Å. Further, it was shown that for FeGe a reduced lattice constant (e.g. by applying high pressures) leads to a non-magnetic semi-conducting ground state as it is found in FeSi 25 .
The spin-polarized eDOS for spin-up and -down are shown in Supplementary Figure 5(b). The ordered magnetic moment is per Fe atom. Optimizing the internal structure of the unit cell for the latter value for the lattice constant, we are able to perform magnetic and non-magnetic calculations and, thus, infer the effect of magnetism on the phonon energies of FeSi. One has to be aware of the fact that the spin-polarized calculation assumes a magnetically ordered state, which is not the case in FeSi at any temperature. However, short-range ferromagnetic correlations between the Fe spins have been discussed in the literature 26 .

Supplementary Note 3 and modetemperature dependences
Spin-polarized calculations predict strong EPC for many phonon modes in FeSi. In the main text we focus on the longitudinal modes at the R point, R1 and R2. Apart from this high symmetry point, Supplementary Figure 8 shows measurements at two other wave vectors, at which we could accurately determine the experimental background and, thereby, were able to extract the phonon line widths as a function of temperature. Two wave vectors are particularly interesting, as DFPT predicts for the mode at q = (0.25, 0.25, 0) and E = 21 meV [arrow in Supplementary Figure 8(a)], whereas the mode in the zone centre at E = 26 meV [arrow in Supplementary Figure 8(d)] should show a line width even larger than that of the R1 and R2 modes. For simplicity, we call them and modes. Panels (b) and (e) show the observed phonon line widths for both phonons. In agreement with DFPT we see no measureable increase of for the mode. The mode, however, clearly features an increasing already below room temperature. For comparison, the temperature dependent of the R1 mode (shifted for the slightly worse calculated resolution for the mode) is indicated by the dashed line in panel (e). Obviously, of the zone centre mode increases already at low temperatures and shows a larger line width at room temperature than the R1 mode. As we discuss in the following in detail, we found that strong EPC apparently goes hand in hand with a particular type of lattice vibrations, in which the Fe-Fe distances are affected by the atomic movements. In order to understand the presence or absence of strongly increased phonon line widths we computed the eigenvectors of the R1 and R2 modes [ Fig. 4 Si ions, we found the following characteristics: For the R1 and R2 modes, the Fe ion located at (u, u, u), u = 0.127 r.l.u., within the unit cell has a displacement nearly parallel to the [111] direction which strongly changes the distance to the other three Fe atoms, which together make roughly a rotational movement around the [111] axis of the unit cell. The pattern of the mode features a pattern with even stronger changes of the inter-Fe distances. Whereas the Fe ion at (u, u, u), u = 0.127 r.l.u., makes a slightly less direct movement towards the other three Fe ions, the latter themselves exhibit a breathing-type pattern, i.e., the three ions move simultaneously towards the center of a circle defined by their equilibrium positions [Supplementary Figure  8(f)]. On the other hand, the mode exhibits evasive movements of the Fe ions, which do not change the interatomic distances very much [ Supplementary Figure 8(c)]. The analysis of the eigenvectors indicates that phonons having patterns with strongly renormalized Fe-Fe distances are affected by additional damping going beyond the quasiharmonic approximation. The effect grows with the temperature-induced paramagnetic moment. Hence, we propose that a direct coupling between the fluctuating electric polarization of the ionic movements and the magnetic moments is responsible for the strong phonon renormalization. FeSi is unique in that it offers to study this effect in a crossover from a non-magnetic insulator towards a nearly ferromagnetic metal, i.e., spanning the magnetic moment range from zero to more than .

Supplementary Note 4 Neutron spin echo measurements
The intrinsic phonon line widths of the R 1 mode at 15 K ≤ T ≤ 225 K were investigated using the three axis spin echo spectrometer TRISP at the neutron research source Hans-Maier Leibnitz (FRM II) in Garching. This instrument applies the neutron spin echo (NSE) technique and is able to determine line widths of the order of a few μeV 27,28 . A bent neutron guide provides a polarized incident neutron beam. The incident energy is selected by a double-focusing graphite monochromator while a velocity selector in front of the monochromator suppresses higher-order energies. Typically, radio-frequency (RF) coils before and after the sample provide the magnetic fields for the neutron precession. Due to the relatively large anharmonic line width of the R 1 mode already at very low temperature, we exchanged the RF coils for DC coils, which enabled us to measure at lower spin-echo times, i.e., τ < 2 ps. We used a fixed final energy of 18.6 meV (|k f |= 3.0 Å -1 ) selected by a Heusler analyzer. Supplementary Figure 3(a) shows the measured polarization for various spin-echo times τ at three selected temperatures. Lines are fits to the data of the form ( ), where is the line width of the Lorentzian line shape describing the intrinsic phonon excitation. Results for are shown in Supplementary Figure 3(b). In order to extract the temperature dependent part of the line width [ Fig. 3(a)], we subtracted the average value of the determined line width for , (blue dashed line). This contribution to the phonon line width originates most likely from anharmonic effects.